We investigate the time series behavior of idiosyncratic volatility and its role in asset pricing in France. We find that both aggregate idiosyncratic and market volatility exhibit regime switching behavior similar to that in the U.S. and other developed countries. Furthermore, we find a marginally significant negative IVOL effect in the French stock market. We add new evidence to the mounting results questioning the ubiquity of the IVOL effect which highlights the importance of country verification of so called anomalies in the US, even in developed markets.
In a recent study, Ang et al. [
Campbell et al. [
In this study, we investigate the behavior of aggregate idiosyncratic and market volatility from 1991 to 2012 in the French stock market. Then we examine the relationship between idiosyncratic volatility and cross-sectional stock returns. There are two reasons why we are interested in the IVOL effect in French stock market. First, most of the previous literature investigates trends and pricing behavior of IVOL in a group of European stock markets, but not in the French stock market on its own. The French stock market is one of the oldest stock markets in the world, and the 2nd largest stock market ranked by capitalization in Europe following the U.K. stock market. Moreover, the French stock exchange was ranked the 4th largest exchange in the world, with a total market capitalization of USD $3.5 trillion in November 2014 [
We can easily summarize our results. First, we test for the presence of trends in aggregate idiosyncratic and market volatility using Bunzel and Vogelsang’s [
We contribute to the literature on the idiosyncratic volatility effect in a number of ways. First, we present the first empirical evidence examining the trend in IVOL and its role in asset pricing in the French stock market. Prior studies mostly focused on a group of developed stock markets rather than on individual stock markets. Hence we draw attention for both academia and practitioners on an individual developed stock market. Second, we add new evidence to the mounting results questioning the ubiquity of the IVOL effect. This highlights the importance of country verification of so called anomalies in the US, even in developed markets. Next, we provide empirical evidence on the pricing of IVOL both at the portfolio and firm levels. Previous research only focuses on the role of pricing IVOL on either portfolio level or firm level. Therefore, our results are comprehensive and more convincing. Finally, we confirm earlier evidence that idiosyncratic volatility in the French stock market exhibits a regime switching behavior rather than showing a long-term time trend.
The rest of the paper is organized as follows: Section 2 describes our data and methods; Section 3 presents the empirical results in three parts. First we report volatility patterns over time, then we examine the relation between volatility and market returns, and finally we examine the relation between idiosyncratic volatility and cross-sectional stock returns. Section 4 concludes the paper.
Although the French stock exchange represents a universal stock exchange in Europe which lists firms from different countries today, the sample of the current study only covers listed French firms. Daily and monthly stock returns on individual firms were obtained from DataStream. The data set covered the period from August 1991 with 147 firms, to June 2012 with 507 firms, with an average of 443 firms per month2. The risk-free rate which is defined as French EU-FRANCE 1 month middle rate was also obtained from the DataStream. Market returns are the value-weighted returns of all firms used in the study.
We exclude investment trusts, closed-end funds, exchange traded funds, and preferred shares. At the beginning of each month, we exclude stocks that do not have continuous return records in the past 22 trading days. In order to reduce noise in computing IVOL for each stock, we also exclude stocks with daily returns less than −100% and/or monthly return greater than 200% as well as stocks with negative book-to-market (BM) ratio. Stocks with missing accounting data in a particular month were also excluded from the sample in that month.
We follow Ang et al. [
R i , t = α + β M K T , i , m M K T t + β S M B , i , m S M B t + β H M L , i , m H M L t + ε i , t (1)
where day t refers to the 22 trading days ending on the last trading day of month m-1. Ri,t and MKT are excess returns of firm i and the market, respectively, over the risk-free rate. SMB is the excess return of small firms over big firms, and HML is the excess return of high book-to-market (BM) firms over low BM firms. SMB is the return of the upper half less the return of the lower half of all firms ranked in ascending order according to market capitalization while HML is the return of the bottom third less the return of the top third of all firms ranked in ascending order according to BM.
We use both portfolio-level analysis as well as firm-level Fama-MacBeth cross-sectional regressions to examine the relation between IVOL and expected returns. In portfolio-level analysis, firms are first sorted into tertiles at the start of each month based on IVOL and allocated to groups. We then compute each tertile portfolio’s equal- and value-weighted raw returns for the current month. We also estimate each tertile portfolio’s alpha (α coefficient) from the FF3-factor model (Equation (1)) using each tertile portfolio’s full sample of monthly value- or equal-weighted returns.
As a robustness test, we also conduct firm-level Fama-MacBeth regressions to control for various variables. We estimate the following model and its nested versions:
R i , t + 1 = β 0 , t + β 1 , t I V i , t + β 2 , t S I Z E i , t + β 3 , t Value i , t + β 4 , t Reversal i , t + β 5 , t Momentum i , t (2)
Rt, is realized stock return in month t. IVOL is realized idiosyncratic volatility as defined previously. SIZE at the end of month t is defined as the log of the firm’s market capitalization at the end of month t. BM is the firm’s book-to-market ratio six months prior, i.e. at the end of t-6. Following Jegadeesh and Titman [
We report the descriptive statistics for three volatility series in
As expected, our volatility measures are highly correlated as shown in Panel B, with correlation coefficients ranging from 0.849 to 0.913.
Panel C displays the autocorrelation structure of the three volatility series. As serial correlation is fairly high in all three series, we test for the presence of unit roots using the augmented Dickey and Fuller [
Panel A: Summary statistics | ||||||
---|---|---|---|---|---|---|
Mean | Median | Stdev | CV | MAX | Min | |
IVOLEW | 0.0188 | 0.0174 | 0.0048 | 0.2553 | 0.0397 | 0.0120 |
IVOLVW | 0.0127 | 0.0113 | 0.0043 | 0.3386 | 0.0322 | 0.0074 |
MVOL | 0.0184 | 0.0165 | 0.0067 | 0.3641 | 0.0582 | 0.0098 |
Panel B: Correlation Table | ||||||
IVOLEW | IVOLVW | MVOL | ||||
IVOLEW | 1.0000 | |||||
IVOLVW | 0.8528 | 1.0000 | ||||
MVOL | 0.8489 | 0.9130 | 1.0000 | |||
Panel C: Autocorrelation structure | ||||||
IVOLEW | IVOLVW | MVOL | ||||
ρ1 | 0.843 | 0.815 | 0.775 | |||
ρ2 | 0.748 | 0.727 | 0.607 | |||
ρ3 | 0.702 | 0.681 | 0.504 | |||
ρ4 | 0.621 | 0.615 | 0.422 | |||
ρ6 | 0.545 | 0.552 | 0.371 | |||
ρ12 | 0.451 | 0.373 | 0.224 | |||
Panel D: Unit root test t-statistics | ||||||
Constant | Constant and Trend | |||||
IVOLEW | −4.6134 | −4.6588 | ||||
IVOLVW | −3.9604 | −3.9728 | ||||
MVOL | −5.5268 | −5.5271 |
Instead, the plots suggest episodic behavior in these series. There was an upward trend in all three volatility series from 1991 to 2001, a downward trend until 2006, a spike in 2009 followed by a decreasing trend thereafter. The pattern of our volatility measures in
As a formal test for the presence of trends, we begin by estimating the following OLS model:
V O L t = b 0 + b 1 t + b 2 V O L t − 1 + ε t (3)
where VOL represents IVOLEW, IVOLVW, and MVOL, and t is time. The estimated time trend b1 parameter and its t-statistic are reported in
1991.08 - 2012.06 | |||
---|---|---|---|
Linear Trend (×10−5) | t-stat | t-dan | |
IVOLEW | 1.6107 | 3.9417 | 1.8160 |
IVOLVW | −2.2946 | −0.6191 | −0.2712 |
MVOL | 0.9200 | 1.5750 | 1.0494 |
In this section, we test for regime-switching behavior in idiosyncratic volatility. We follow Bekaert et al.’s [
To test for regime switching behavior in idiosyncratic volatility in the French stock market, we let volatility, yt, follow an AR(1) model where all parameters can take on one of two values depending on the realization of a discrete regime variable, st. The regime variable follows a Markov chain with constant transition probabilities. Indexing the current regime by i the model is
y t = ( 1 − b i ) μ i + b i y t − 1 + σ i e t , i ∈ { 0 , 1 } (4)
with et ~ N (0,1). In the model, we force regime 0 (regime 1) to be the low (high) volatility regime and the mean levels (μi) of the volatility series of both regimes to be nonnegative (i.e. μ1 > μ0 > 0).
The transition probability matrix, Φ, is 2 × 2, where each probability represents P [ s t = i | s t − 1 = j ] , with i , j ∈ { 1 , 2 } :
Φ = ( p 11 1 − p 11 1 − p 22 p 22 )
The model has only 8 parameters, { μ 0 , μ 1 , b 0 , b 1 , σ 0 , σ 1 , p 11 , p 22 } .
The estimation results for each volatility series (yt = IVOLEW, IVOLVW, and MVOL) are reported in
IVOLEW | IVOLVW | MVOL | ||||
---|---|---|---|---|---|---|
Coeff. | Stan. Error | Coeff. | Stan. Error | Coeff. | Stan. Error | |
p11 | 0.9566 (51.9802) | 0.0184 | 0.9359 (20.4997) | 0.0457 | 0.5898 (5.3824) | 0.1096 |
p22 | 0.8497 (11.4633) | 0.0741 | 0.9762 (58.2982) | 0.0167 | 0.8974 (28.3458) | 0.0317 |
σ0 | 0.0013 (14.8685) | 0.0001 | 0.0013 (16.4254) | 0.0001 | 0.0018 (14.7378) | 0.0001 |
σ1 | 0.0040 (9.4579) | 0.0004 | 0.0035 (11.7100) | 0.0003 | 0.0064 (9.4864) | 0.0007 |
b0 | 0.4815 (3.6061) | 0.1335 | 0.6497 (11.0250) | 0.0589 | 0.6357 (26.9328) | 0.0236 |
b1 | 0.7600 (19.2827) | 0.0394 | 0.4810 (4.2974) | 0.1119 | 0.6902 (3.3680) | 0.2049 |
μ0 | 0.0258 (20.1068) | 0.0013 | 0.0105 (36.1627) | 0.0003 | 0.0147 (34.8796) | 0.0004 |
μ1 | 0.0163 (36.9071) | 0.0004 | 0.0181 (19.0951) | 0.0010 | 0.0355 (4.0839) | 0.0087 |
Likelihood | −1205.3056 | −1216.8193 | −1103.8582 |
to compute the standard errors.
volatility regime from 1992 to mid-2000, consistent with Morel’s [
We also find it interesting that IVOLVW and IVOLEW exhibit a divergence in the period from 1992 to 1999, with IVOLEW being on a high-volatility regime while IVOLVW was on a low-volatility regime. We suggest that this could be due to the boom in high-tech stocks over this period. As high-tech stocks are normally smaller in size and more volatile than traditional listed firms, we expect IVOLEW to be more volatile than IVOLVW before the high-tech bubble burst around year 2000. We also find that both IVOLVW and IVOLEW were on a high volatility regime during the recent 2008 financial crisis. This is consistent with previous findings in the literature wherein stock markets are more volatile during the financial crisis period than other periods [
In sum, the results from
In this section we examine the presence of an IVOL effect in the French stock market.
Before we test the robustness of this apparent negative IVOL effect in the French stock market, we report the average of the monthly averages of various
Raw Return | FF-3 Alpha | |||||
---|---|---|---|---|---|---|
Mean | Std. Dev | Mean | Std. Error | |||
Equal-weighted | ||||||
High IVOL | −0.0030 (−0.6170) | 0.0059 | −0.0003 (−1.1118) | 0.0026 | ||
Medium IVOL | 0.0027 (0.7707) | 0.0030 | 0.0010 (0.5384) | 0.0018 | ||
Low IVOL | 0.0061 (2.3879) | 0.0016 | 0.0031 (2.3995) | 0.0013 | ||
High-Low | −0.0091 (−1.6526) | −0.0034 (−1.1696) | 0.0029 | |||
Value-weighted | ||||||
High IVOL | −0.0019 (−0.3658) | 0.0070 | −0.0039 (−2.0721) | 0.0019 | ||
Medium IVOL | 0.0028 (0.7288) | 0.0038 | −0.0008 (−0.7252) | 0.0011 | ||
Low IVOL | 0.0051 (1.5867) | 0.0026 | 0.0019 (3.4614) | 0.0006 | ||
High-Low | −0.0071 (−1.1363) | −0.0058 (−2.9109) | 0.0020 | |||
characteristics of the IVOL-sorted portfolios in
We begin with univariate regressions on IVOL and our control variables.
IVOL | SIZE | Value | Momentum | REV | |
---|---|---|---|---|---|
High IVOL | 0.0303 (55.7154) | 575.82 (23.9111) | 0.8218 (37.8627) | −0.0394 (−1.5912) | 0.0044 (0.7793) |
Medium IVOL | 0.0166 (61.3300) | 1810.59 (32.4654) | 0.7864 (48.2870) | 0.0638 (4.1401) | 0.0022 (0.6770) |
Low IVOL | 0.0095 (69.2589) | 3717.68 (33.1339) | 0.8385 (76.4216) | 0.0924 (8.9538) | 0.0004 (0.2133) |
High-Low | 0.0208 (37.1756) | −3141.86 (−27.3784) | −0.0200 (−0.8192) | −0.1318 (−4.9116) | 0.0040 (0.6716) |
Intercept | IVOL | SIZE | Value | Reversal | Momentum |
---|---|---|---|---|---|
0.0117 (3.10) | −0.6322 (−3.08) | ||||
0.0017 (0.48) | 1.16E−7 (0.68) | ||||
−0.0035 (−0.72) | 0.0072 (3.75) | ||||
0.0022 (0.53) | −0.0135 (−1.02) | ||||
0.0005 (0.15) | 0.0159 (4.21) |
show significant momentum and BM effects with previous winners and stocks with high BM exhibiting higher returns. However we find no size and short-term reversal effects. The absence of a size effect is consistent with Morel [
Intercept | IVOL | SIZE | Value | Reversal | Momentum |
---|---|---|---|---|---|
0.0130 (3.03) | −0.7141 (−2.92) | −5.93E−8 (−0.35) | |||
0.0056 (1.52) | −0.5235 (−3.14) | 0.0063 (3.99) | |||
0.0098 (2.83) | −0.4504 (−3.12) | −0.0164 (−1.39) | |||
0.0093 (2.35) | −0.5830 (−2.82) | 0.0142 (3.87) | |||
0.0021 (0.49) | −0.4454 (−1.75) | 4.97E−9 (0.03) | 0.0063 (4.99) | −0.0495 (−3.68) | 0.0210 (5.07) |
Next we control the size, BM, reversal, and momentum effects individually with bi-variate Fama-MacBeth cross-sectional regressions and then simultaneously in a multivariate regression. We report the results in
In a recent study, Ang et al. [
We find that both idiosyncratic and market volatility do not exhibit long-term trends. Instead their patterns are consistent with regime switching behavior similar to that in the U.S. and other developed countries. Though we initially find a negative IVOL effect in the French stock market which is robust in bi-variate Fama-MacBeth regressions, the negative IVOL effect promptly disappears when we control for these well-known effects simultaneously.
We add new evidence to the mounting results questioning the ubiquity of the IVOL effect which highlights the importance of country verification of so-called anomalies in the US, even in developed markets.
We thank two anonymous reviewers for their constructive comments. We also wish to thank the participants of the World Business, Finance, and Management Conference in December, 2014, Auckland, New Zealand for their comments and suggestions on earlier versions of this paper.
Liu, Z.T., Nartea, G.V. and Wu, J. (2018) Patterns and Pricing of Idiosyncratic Volatility in the French Stock Market. Theoretical Economics Letters, 8, 79-97. https://doi.org/10.4236/tel.2018.81005
Appendix 1. Description of the sample stocks.